Adding up to 6

Number Theory Level 3

Using more than one positive integer, how many different ways are there to add up to $6$ ?

Note: Do not count combinations like $2+4$ and $4+2$ as different.

3 4 8 10

1 solution

Jayaditya Venkatesan
Nov 12, 2020

There can be two-numbered expressions ( $2+4$ ), three-numbered expressions ( $2+2+2$ ), four-numbered expressions ( $1+1+1+3$ ), five-numbered expressions $(1+1+1+1+2$ ), and a six-numbered expression $(1+1+1+1+1+1$ ).

Two-numbered expressions: $2+4$ , $3+3$ , and $5+1$ .
Three-numbered expressions: $1+2+3$ , $1+1+4$ , and $2+2+2$ .
Four-numbered expressions: $1+1+1+3$ and $1+1+2+2$ .
Five-numbered expressions: $1+1+1+1+2$ .
Six-numbered expressions: $1+1+1+1+1+1$ .

In total, there are $\boxed{10}$ ways to add up to $6$ .

Adding up to 6

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